Slicer can display anything that you see in the 3D viewer in a virtual reality headset directly, using SteamVR, without Unity. Unity is really limited when it comes to medical image visualization: you have to work hard to get such basic features as reslicing volumes, volume rendering, and visualization of meshes that consists of more than a few hundred thousand points.
We only use Unity in connection with Slicer when we run standalone applications on Microsoft HoloLens - in that case Slicer sends transforms, images, and surface meshes through OpenIGTLink. However, probably this is not what you are looking for.
Currently, we find that moving your head inside organs works very well as a “dynamic clipping plane”, but of course having an additional independent cutting can be very useful.
Slicer’s virtual reality extension has already a built-in cutting plane feature. You can change controller mode by clicking one of the buttons on the controller then tilting the controller to choose mode. However, this is still experimental (you can only clip meshes, they are only clipped in the virtual reality view, etc.) but it’s there, so you can try it and maybe improve it.